Simplify the following expression and state the condition under which the simplification is valid. $y = \dfrac{r^3 - 9r}{-6r^2 - 42r - 72}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ y = \dfrac {r(r^2 - 9)} {-6(r^2 + 7r + 12)} $ $ y = -\dfrac{r}{6} \cdot \dfrac{r^2 - 9}{r^2 + 7r + 12} $ Next factor the numerator and denominator. $ y = - \dfrac{r}{6} \cdot \dfrac{(r + 3)(r - 3)}{(r + 3)(r + 4)}$ Assuming $r \neq -3$ , we can cancel the $r + 3$ $ y = - \dfrac{r}{6} \cdot \dfrac{r - 3}{r + 4}$ Therefore: $ y = \dfrac{ -r(r - 3)}{ 6(r + 4)}$, $r \neq -3$